In the baseball World Series, two teams play games until one...

In the baseball World Series, two teams play games until one team or the other wins four times. Suppose team A should win each game with probability 0.6. The probabilities for team A's record (given as wins/losses) in the World Series are shown. Find the probability of each event indicated below. $$\begin{array}{|l|l|l|l|l|} \hline \text { Wins/losses } & 0 / 4 & 1 / 4 & 2 / 4 & 3 / 4 \\ \hline \text { Probability } & 0.0256 & 0.0615 & 0.0922 & 0.1106 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|} \hline \text { Winshosses } & 4 / 3 & 4 / 2 & 4 / 1 & 4 / 0 \\ \hline \text { Probability } & 0.1659 & 0.2073 & 0.2073 & 0.1296 \\ \hline \end{array}$$ (a) Team A wins the World Serics (b) Team B wins the World Series (c) One team wins all four games (d) The teams play six or seven games

In the baseball World Series, two teams play games until one team or the other wins four times. Suppose team A should win each game with probability 0.6. The probabilities for team A's record (given as wins/losses) in the World Series are shown. Find the probability of each event indicated below.
$$\begin{array}{|l|l|l|l|l|}
\hline \text { Wins/losses } & 0 / 4 & 1 / 4 & 2 / 4 & 3 / 4 \\
\hline \text { Probability } & 0.0256 & 0.0615 & 0.0922 & 0.1106 \\
\hline
\end{array}$$
$$\begin{array}{|l|l|l|l|l|}
\hline \text { Winshosses } & 4 / 3 & 4 / 2 & 4 / 1 & 4 / 0 \\
\hline \text { Probability } & 0.1659 & 0.2073 & 0.2073 & 0.1296 \\
\hline
\end{array}$$
(a) Team A wins the World Serics
(b) Team B wins the World Series
(c) One team wins all four games
(d) The teams play six or seven games

Key Concepts

Best-of-Seven Series Structure

This concept refers to the competitive format where two teams play a maximum of seven games, and the series ends as soon as one team wins four games. It introduces the idea of an absorbing state in a sequential game process, where the series terminates once a decisive outcome is reached.

Independent Game Probabilities

Each game in the series is treated as an independent event, with its own fixed probability of one team winning. This assumption underlies the computation of overall series outcomes and simplifies the analysis by allowing the multiplication of individual game probabilities.

Combinatorial Analysis of Sequential Outcomes

This involves counting the number of different orders in which wins and losses can occur before a series reaches the four-win threshold. Since the series stops immediately once a team collects four wins, combinatorial techniques are adapted to account for these early terminations in calculating overall probabilities.

Truncated Binomial (or Negative Binomial) Distribution

The series does not necessarily play all seven games, so the usual binomial distribution must be adjusted or truncated. This concept deals with distributions where the number of trials is not fixed because the process stops once a predetermined number of successes is achieved, requiring conditional probability methods.

Sweep in a Series

A 'sweep' occurs when one team wins all the necessary games in succession without allowing the opponent any wins. Calculating the probability of a sweep involves raising the single-game win probability to the power corresponding to the number of consecutive wins needed, reflecting the compounding nature of independent events.

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